Odds, percentage, chances, etc... are simply too complicated to determine. As already has been mentioned, it is totally based on your competition. You compete at the district level of your state, as well as possibly at the national level. There's also "other" areas where you may be competing, such as against those with a presidential, ROTC, VP, etc... nomination.
Using the most simplest example. There are 435 US representatives. There are 100 senators. That's 535 slates with 10 names each; = 5350 Nominees. We'll take out 25% for those who have more than one MOC nomination, and we're in the ball park of approximately 4,000 individuals with nominations. YET.... there are only between 1,000 and 1,200 actual appointments.
To complicate it a little more, of those 1,000 - 1,200 appointments, approximately 100 are going to be from those with a Presidential nomination. Another 20 will be from those with an ROTC nomination. Another 50+ will be prior enlisted. Then, there's the other misc nominations. So, for the sake of argument, let's say there's 950-975 for those with MOC nominations.
Now, assuming that ALL 535 MOCs had a slot, and thus submitted a slate of 10 names, the academy, whether they like it or not, MUST give one appointment per slate. (There's 3 different ways that appointment can be decided, but that's no important here). What's important is; if YOU are the BEST and #1 rated on your MOC's list, then you probably have nothing to worry about. But for the sake of argument, let's assume you aren't the #1 best applicant on your MOC's list. That means, another 535 appointments out of 950-975 we narrowed down to, have been given out. That leaves approximately 450 appointments left to give out.
Now remember; not counting the presidentials, vp, etc... we came up with approximately 4,000 individuals with MOC nominations. We'll even shorten that list down and say that 1,000 of those individuals are somehow not qualified. E.g. medically, CFA, academic. That leaves 3,000. Minus the 535 appointments that we already gave out, that leave 2,465 individuals competing over 450 remaining appointments. Which means:
ASSUMING you are medically, physically, and academically qualified:
1. If you are the BEST and #1 on your MOC's slate, then you have a 90% chance of an appointment. (The reason it's not 100% is because of variables such as being on more than 1 list and someone ahead of you turning down the appointment). But we are assuming that you are the BEST on the slate.
2. If you aren't the BEST on the slate, and you must compete in the national pool, then you have an 18.25% chance of receiving an appointment.
FWIW: Because it totally depends on your competition, and no one knows your competition, the 18.25% number implies that ALL applicants in the national pool are equal in their application. The truth is, you could have such a strong application vs the other 2,465 individuals, that your chances could be at 97.38%. On the other hand, you migh have been the #10 on your MOC's slate, and number $2460 on the national list, giving you a 1.34% chance. Statistics SUCK. Best of luck to you.